One of the most interesting problems of geometry, the solution to which is important in physics, chemistry, and other fields, is the determination of volumes. While doing math in school, children often ask themselves: “Why do we need this?” The world around it seems so simple and understandable that certain school knowledge is classified as "unnecessary." But it is worthwhile to encounter, for example, transportation and the question arises of how to calculate the volume of cargo. Say nothing is easier? You are mistaken. Knowledge of design formulas, the concepts of "density of matter", "bulk density of bodies" are becoming necessary.
School knowledge - a practical basis
Teachers of schools, teaching the basics of geometry, offer us this definition of volume: the part of the space occupied by the body. Moreover, the formulas for determining volumes have long been recorded, and you can find them in the reference books. Mankind learned to determine the volume of a body of the correct form long before the advent of the treatises of Archimedes. But only this great Greek thinker introduced a methodology that makes it possible to determine the volume of any figure. His conclusions became the basis of integral calculus. Volumetric figures are considered to be obtained in the process of rotation of flat geometric figures.
Euclidean geometry with a certain accuracy allows to determine the volume:
Geometric body | Calculation formula | main parameters |
Rectangular box | V = lbh | l - length, b - width, h - height |
Cube | V = a 3 | a - cube edge |
Cylinder | V = sh | S - base area, h - height |
Sphere | V = 4πR 3/3 | R is the radius of the sphere |
The difference between flat and three-dimensional figures does not allow us to answer the question of some sufferers about how to calculate the volume of a rectangle. It’s about the same as finding something, I don’t know what. Confusion in the geometric material is possible, while a rectangle is sometimes called a rectangular box.
What to do if the shape of the body is not so clearly defined?
Determining the volume of complex geometric structures is not an easy job. It should be guided by several unshakable principles.
- Any body can be divided into simpler parts. The volume is equal to the sum of the volumes of its individual parts.
- Equal-sized bodies have equal volumes, parallel transfer of bodies does not change its volume.
- The unit of volume is the volume of a cube with an edge of unit length.
The presence of bodies of irregular shape (recall the notorious crown of King Heron) does not become a problem. The determination of body volumes by hydrostatic weighing is quite possible. This is the process of directly measuring the volume of liquid with a body immersed in it, which will be discussed below.
Various scope applications
Let's return to the problem: how to calculate the volume of transported goods. What is the cargo: packaged or loose? What are the packaging options? There are more questions than answers. An important question will be the mass of the cargo, since transport is characterized by carrying capacity, and routes - by the maximum weight of the vehicle. Violation of the rules of transportation threatens with penalties.
Task 1. Let the cargo be rectangular containers filled with goods. Knowing the weight of the product and container, you can easily determine the total weight. The volume of the container is defined as the volume of the rectangular box.
Knowing the carrying capacity of the vehicle, its dimensions, you can calculate the possible volume of transported cargo. The correct ratio of these parameters allows you to avoid disaster, premature failure of transport.
Task 2. Cargo - bulk material: sand, gravel and the like. At this stage, only a classy specialist can do without knowledge of physics, whose experience in cargo transportation allows you to intuitively determine the maximum permissible volume for transportation.
The scientific method involves the knowledge of such a parameter as the density (bulk density) of the cargo.
The formula V = m / ρ is used, where m is the mass of the load, ρ is the density of the material. Before calculating the volume, it is worth knowing the density of the cargo, which is also not at all difficult (tables, laboratory definition).
This technique also works great in determining liquid cargo volumes. Moreover, a liter is used as a unit of measure.
Definition of volumes of building forms
The issue of determining volumes plays an important role in construction. The construction of houses and other structures is expensive, building materials require careful attention and extremely accurate calculation.
The foundation of the building - the foundation - is usually a cast structure filled with concrete. Before calculating the volume of concrete, it is necessary to determine the type of foundation.
The slab foundation is a slab in the form of a rectangular parallelepiped. Columnar base - rectangular or cylindrical pillars of a certain section. Having determined the volume of one column and multiplying it by the number, you can calculate the cubic capacity of concrete for the entire foundation.
Calculating the volume of concrete for walls or ceilings, they do it quite simply: they determine the volume of the entire wall by multiplying the length by the width and height, then separately determine the volumes of window and door openings. The difference between the volume of the wall and the total volume of openings is the volume of concrete.
How to determine the volume of a building?
Some applications require knowledge about the volume of buildings and structures. These include problems of repair, reconstruction, determination of air humidity, issues related to heat supply and ventilation.
Before answering the question of how to calculate the volume of a building, take measurements on its outer side: sectional area (length multiplied by width), building height from the bottom of the first floor to the attic.
The determination of the internal volumes of heated rooms is carried out according to the internal strokes.
The device of heating systems
Modern apartments and offices cannot be imagined without a heating system. The main part of the systems are batteries and connecting pipes. How to calculate the volume of the heating system? The total volume of all heating sections, which is indicated on the radiator itself, must be added to the volume of pipes.
And at this stage the problem arises: how to calculate the volume of the pipe. Imagine that a pipe is a cylinder, the solution comes by itself: we use the formula for calculating the volume of the cylinder. In heating systems, pipes are filled with water, so you need to know the internal cross-sectional area of the pipe. To do this, determine its internal radius (R). The formula for determining the area of a circle: S = πR 2 . The total length of the pipes is determined by their length in the room.
Sewerage in the house - pipe system
When laying pipes for drainage, it is also worth knowing the volume of the pipe. At this stage, an external diameter is required, the actions are similar to the previous ones.
Determining the volume of metal used to make the pipe is also an interesting task. Geometrically, a pipe is a cylinder with voids. Determining the area of the ring lying in its cross section is a rather complicated but solvable task. A simpler solution is to determine the external and internal volumes of the pipe, the difference between these values will be the volume of the metal.
Volume determination in physics problems
The famous legend of the crown of Tsar Geron became known not only as a result of solving the problem of bringing thieving jewelers “into the clear water”. The result of the complex mental activity of Archimedes is the determination of the volumes of bodies of irregular geometric shapes. The main idea extracted by the philosopher is that the volume of fluid displaced by the body is equal to the volume of the body.
In laboratory studies, use a measuring cylinder (beaker). Determine the volume of liquid (V 1 ), immerse the body in it, perform secondary measurements (V 2 ). The volume is equal to the difference between the secondary and primary measurements: V t = V 2 - V 1 .
This method of determining body volumes is used in calculating the bulk density of bulk insoluble materials. It is extremely convenient when determining the density of alloys.
The pin volume can be calculated using this method. It seems quite difficult to determine the volume of such a small body as a pin or pellet. It cannot be measured with a ruler; a graduated cylinder is also large enough.
But if you use several completely identical pins (n), then you can use the measuring cylinder to determine their total volume (V t = V 2 - V 1) . Then divide the resulting value by the number of pins. V = V t \ n.
This task becomes clear if many pellets must be cast from one large piece of lead.
Fluid Volume Units
The international system of units involves the measurement of volumes in m
3 . In everyday life, extra-systemic units are often used: liter, milliliter. When determining how to calculate the volume in liters, use the translation system: 1 m
3 = 1000 liters.
The use of other non-systemic measures in everyday life can cause difficulties. The British use more familiar barrels, gallons, and bushels.
Translation System:
English measures | Russian measures |
Bushel | 36.4 l | Bucket | 12 l |
Gallon | 4,5 l | Barrel | 490 l |
Barrel (dry) | 115.628 l | Damask | 1, 23 l |
Barrel (oil) | 158, 988 l | Charka | 0, 123 l |
English barrel for bulk solids | 163.65 L | Shkalik | 0.06 L |
Custom Data Tasks
Task 1. How to calculate the volume, knowing the height and area? Usually this problem is solved by determining the amount of coating of various parts by galvanic means. Moreover, the surface area of the part (S) is known. Layer thickness (h) - height. Volume is determined by the product of area and height: V = Sh.
Problem 2. For cubes, interesting, from a mathematical point of view, the task of determining the volume may look if the area of one face is known. It is known that the volume of the cube: V = a 3 , where a is the length of its face. The lateral surface area of the cube is S = a 2 . Extracting the square root from the area, we get the length of the face of the cube. We use the volume formula, calculate its value.
Problem 3. Calculate the volume of the figure, if the area is known and some parameters are given. Additional parameters include conditions for the aspect ratio, heights, base diameters, and much more.
To solve specific problems, you will need not only knowledge of volume calculation formulas, but also other geometry formulas.
Determining the amount of memory
A task completely unrelated to geometry: to determine the memory capacity of electronic devices. In the modern, fairly computerized world, this problem is not unnecessary. Exact devices, such as personal computers, do not tolerate approximation.
Knowing the memory capacity of a flash drive or other storage device is useful when copying, moving information.
It is important to know the amount of operational and permanent memory of the computer. Often the user is faced with a situation where "the game is not running", "the program hangs." The problem is quite possible with low memory.
A unit of information is considered a byte and its derivatives (kilobytes, megabytes, terabytes).
1 kB = 1024 B
1 MB = 1024 kB
1 GB = 1024 MB
Strangeness in this recalculation system follows from the binary information coding system.
The memory size of the storage device is its main characteristic. By comparing the amount of information transferred and the amount of storage memory, you can determine the possibility of its further operation.
The concept of “volume” is so large-scale that one can fully understand its versatility only by solving applied problems, interesting and fascinating.